May a layperson baptize in emergency?
Your question should be addressed to a Roman Catholic.
Presbyterians believe the God who created that person who may be dying "in emergency" already knows the heart of that dying person since it was God's gift in the first place.
As a nurse, my mother baptized several dying infants, but only at the request of the child's grieving Roman Catholic parents.
A better security for our children is for those children to have two believing parents who know that everything comes from the Lord for their benefit, one way or another, by grace through faith in Jesus Christ.
"For the promise is unto you, and to your children, and to all that are afar off, even as many as the LORD our God shall call." -- Acts 2:39
I can't conceive of a need for an emergency (water) baptism...
After Philip led the Ethiopian to Jesus, the Ethiopian was immediately filled with the Holy Spirit (spiritual baptism)...Philip never asked or suggested the Ethiopian needed to get wet for his salvation to kick in...And Philip may never have baptized the Ethiopian had the Ethiopian not asked for it...